Theorem rnmpt 5010

Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
rnmpt	⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem rnmpt

Step	Hyp	Ref	Expression
1		rnopab 5009	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		rnmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		df-mpt 4178	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	2, 3	eqtri 2255	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5	4	rneqi 4990	. 2 ⊢ ran 𝐹 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6		df-rex 2528	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
7	6	abbii 2350	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
8	1, 5, 7	3eqtr4i 2265	1 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∃wrex 2523  {copab 4175   ↦ cmpt 4176  ran crn 4755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-mpt 4178  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  elrnmpt  5011  elrnmpt1  5013  elrnmptg  5014  dfiun3g  5019  dfiin3g  5020  fnrnfv  5728  fmpt  5832  fnasrn  5861  fnasrng  5863  fliftf  5978  abrexex  6319  abrexexg  6320  fo1st  6364  fo2nd  6365  qliftf  6867  negfi  11938  4sqlem11  13124  4sqlem12  13125  quslem  13588  restco  15165  2lgslem1b  16088